Steiner quadruple systems with point-regular abelian automorphism groups

TL;DR

This paper introduces a graph-theoretic method to construct Steiner quadruple systems with abelian automorphism groups, focusing on A-reversibility and providing existence conditions for specific group structures.

Contribution

It presents a novel construction approach for A-reversible Steiner quadruple systems and establishes existence criteria based on the group's Sylow 2-subgroup properties.

Findings

A-reversible SQS always exists when A is a 2-group of exponent at most 4.

Necessary and sufficient conditions are given for A-reversible SQS when the Sylow 2-subgroup of A is cyclic.

Construction of A-reversible SQS is possible for any abelian group A with certain dihedral SQS conditions.

Abstract

In this paper we present a graph theoretic construction of Steiner quadruple systems (SQS) admitting abelian groups as point-regular automorphism groups. The resulting SQS has an extra property which we call A-reversibility, where A is the underlying abelian group. In particular, when A is a 2-group of exponent at most 4, it is shown that an A-reversible SQS always exists. When the Sylow 2-subgroup of A is cyclic, we give a necessary and sufficient condition for the existence of an A-reversible SQS, which is a generalization of a necessary and sufficient condition for the existence of a dihedral SQS by Piotrowski (1985). This enables one to construct A-reversible SQS for any abelian group A of order v such that for every prime divisor p of v there exists a dihedral SQS(2p).